Group (mathematics)/Catalogs
From Knowino
The mathematical group concept represents a rather simple and natural generalization of common phenomena, so examples of groups are easily found, from all areas of mathematics.
[edit] Different classes of groups
Three different classes of groups are commonly studied:
[edit] Examples of finite discrete groups
- The trivial group consisting of just one element.
- The group of order two, which f.i. can be represented by addition modulo 2 or the set {-1, 1} under multiplication.
- The group of order three.
- The cyclic group of order 4, which can be represented by addition modulo 4.
- The noncyclic group of order 4, known as the "Klein viergruppe". A simple physical model of this group is two separate on-off switches.
[edit] Some physical models
Some common physical objects provide excellent introductions to group theory.
Model of the cyclic group of order 4.
It's easy to see the following:
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Model of the non-cyclic group of order 4.
It's easy to see the following:
These results can be summarized in the following table:
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Many examples of groups come from considering some object and a set of bijective functions from the object to itself, which preserve some structure that this object has.
- Topological groups:
- Matrix groups; e.g the general linear group, the special linear group, the projective linear group and the Platonic groups
- Abelian varieties.
- Finite groups.
- Galois groups.
- Fundamental groups.